text: //=================================================================================================
  /*!
  //  \file blaze/math/dense/LQ.h
  //  \brief Header file for the dense matrix in-place LQ decomposition
  //
  //  Copyright (C) 2013 Klaus Iglberger - All Rights Reserved
  //
  //  This file is part of the Blaze library. You can redistribute it and/or modify it under
  //  the terms of the New (Revised) BSD License. Redistribution and use in source and binary
  //  forms, with or without modification, are permitted provided that the following conditions
  //  are met:
  //
  //  1. Redistributions of source code must retain the above copyright notice, this list of
  //     conditions and the following disclaimer.
  //  2. Redistributions in binary form must reproduce the above copyright notice, this list
  //     of conditions and the following disclaimer in the documentation and/or other materials
  //     provided with the distribution.
  //  3. Neither the names of the Blaze development group nor the names of its contributors
  //     may be used to endorse or promote products derived from this software without specific
  //     prior written permission.
  //
  //  THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" AND ANY
  //  EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES
  //  OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT
  //  SHALL THE COPYRIGHT HOLDER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT,
  //  INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED
  //  TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR
  //  BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN
  //  CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN
  //  ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH
  //  DAMAGE.
  */
  //=================================================================================================
  
  #ifndef _BLAZE_MATH_DENSE_LQ_H_
  #define _BLAZE_MATH_DENSE_LQ_H_
  
  
  //*************************************************************************************************
  // Includes
  //*************************************************************************************************
  
  #include <memory>
  #include <blaze/math/Aliases.h>
  #include <blaze/math/constraints/Adaptor.h>
  #include <blaze/math/constraints/BLASCompatible.h>
  #include <blaze/math/constraints/Hermitian.h>
  #include <blaze/math/constraints/StrictlyTriangular.h>
  #include <blaze/math/constraints/Symmetric.h>
  #include <blaze/math/constraints/UniTriangular.h>
  #include <blaze/math/constraints/Upper.h>
  #include <blaze/math/Exception.h>
  #include <blaze/math/expressions/DenseMatrix.h>
  #include <blaze/math/lapack/gelqf.h>
  #include <blaze/math/lapack/orglq.h>
  #include <blaze/math/lapack/unglq.h>
  #include <blaze/math/traits/DerestrictTrait.h>
  #include <blaze/math/typetraits/IsResizable.h>
  #include <blaze/math/typetraits/IsSquare.h>
  #include <blaze/math/views/Submatrix.h>
  #include <blaze/util/algorithms/Min.h>
  #include <blaze/util/EnableIf.h>
  
  
  namespace blaze {
  
  //=================================================================================================
  //
  //  LQ DECOMPOSITION FUNCTIONS
  //
  //=================================================================================================
  
  //*************************************************************************************************
  /*!\name LQ decomposition functions */
  //@{
  template< typename MT1, bool SO1, typename MT2, bool SO2, typename MT3, bool SO3 >
  void lq( const DenseMatrix<MT1,SO1>& A, DenseMatrix<MT2,SO2>& Q, DenseMatrix<MT3,SO3>& R );
  //@}
  //*************************************************************************************************
  
  
  //*************************************************************************************************
  /*! \cond BLAZE_INTERNAL */
  /*!\brief Auxiliary function for the LQ decomposition.
  // \ingroup dense_matrix
  //
  // \param A The LQ decomposed column-major matrix.
  // \param tau Array for the scalar factors of the elementary reflectors.
  // \return void
  //
  // This function is an auxiliary helper for the dense matrix LQ decomposition. It performs the
  // reconstruction of the \c Q matrix from the RQ decomposition.
  */
  template< typename MT1 >  // Type of matrix A
  inline EnableIf_<IsBuiltin< ElementType_<MT1> > >
     lq_backend( MT1& A, const ElementType_<MT1>* tau )
  {
     orglq( A, tau );
  }
  /*! \endcond */
  //*************************************************************************************************
  
  
  //*************************************************************************************************
  /*! \cond BLAZE_INTERNAL */
  /*!\brief Auxiliary function for the LQ decomposition.
  // \ingroup dense_matrix
  //
  // \param A The LQ decomposed column-major matrix.
  // \param tau Array for the scalar factors of the elementary reflectors.
  // \return void
  //
  // This function is an auxiliary helper for the dense matrix LQ decomposition. It performs the
  // reconstruction of the \c Q matrix from the RQ decomposition.
  */
  template< typename MT1 >  // Type of matrix A
  inline EnableIf_<IsComplex< ElementType_<MT1> > >
     lq_backend( MT1& A, const ElementType_<MT1>* tau )
  {
     unglq( A, tau );
  }
  /*! \endcond */
  //*************************************************************************************************
  
  
  //*************************************************************************************************
  /*!\brief LQ decomposition of the given dense matrix.
  // \ingroup dense_matrix
  //
  // \param A The matrix to be decomposed.
  // \param L The resulting \c L matrix.
  // \param Q The resulting \c Q matrix.
  // \return void
  // \exception std::invalid_argument Dimensions of fixed size matrix do not match.
  // \exception std::invalid_argument Square matrix cannot be resized to \a m-by-\a n.
  //
  // This function performs the dense matrix LQ decomposition of a general \a m-by-\a n matrix.
  // The resulting decomposition has the form
  
                                \f[ A = L \cdot Q, \f]
  
  // where \c L is a lower trapezoidal \a m-by-min(\a m,\a n) matrix and \c Q is a general
  // min(\a m,\a n)-by-\a n matrix. The decomposition is written to the two distinct matrices
  // \c L and \c Q, which are resized to the correct dimensions (if possible and necessary).
  //
  // The function fails if ...
  //
  //  - ... either \a L or \a Q are fixed size matrices and the dimensions don't match;
  //  - ... \a L is a compile time square matrix, but is required to be non-square.
  //
  // In all failure cases a \a std::invalid_argument exception is thrown.
  //
  // Example:
  
     \code
     blaze::DynamicMatrix<double,blaze::columnMajor> A( 32, 16 );
     // ... Initialization of A
  
     blaze::DynamicMatrix<double,blaze::columnMajor> L( 32, 16 );
     blaze::DynamicMatrix<double,blaze::columnMajor> Q( 16, 16 );
  
     lq( A, L, Q );
  
     assert( A == L * Q );
     \endcode
  
  // \note This function only works for matrices with \c float, \c double, \c complex<float>, or
  // \c complex<double> element type. The attempt to call the function with matrices of any other
  // element type results in a compile time error!
  //
  // \note This function can only be used if a fitting LAPACK library is available and linked to
  // the executable. Otherwise a call to this function will result in a linker error.
  */
  template< typename MT1  // Type of matrix A
          , bool SO1      // Storage order of matrix A
          , typename MT2  // Type of matrix L
          , bool SO2      // Storage order of matrix L
          , typename MT3  // Type of matrix Q
          , bool SO3 >    // Storage order of matrix Q
  void lq( const DenseMatrix<MT1,SO1>& A, DenseMatrix<MT2,SO2>& L, DenseMatrix<MT3,SO3>& Q )
  {
     BLAZE_CONSTRAINT_MUST_NOT_BE_STRICTLY_TRIANGULAR_MATRIX_TYPE( MT1 );
     BLAZE_CONSTRAINT_MUST_BE_BLAS_COMPATIBLE_TYPE( ElementType_<MT1> );
  
     BLAZE_CONSTRAINT_MUST_NOT_BE_SYMMETRIC_MATRIX_TYPE( MT2 );
     BLAZE_CONSTRAINT_MUST_NOT_BE_HERMITIAN_MATRIX_TYPE( MT2 );
     BLAZE_CONSTRAINT_MUST_NOT_BE_UNITRIANGULAR_MATRIX_TYPE( MT2 );
     BLAZE_CONSTRAINT_MUST_NOT_BE_UPPER_MATRIX_TYPE( MT2 );
     BLAZE_CONSTRAINT_MUST_BE_BLAS_COMPATIBLE_TYPE( ElementType_<MT2> );
  
     BLAZE_CONSTRAINT_MUST_NOT_BE_ADAPTOR_TYPE( MT3 );
     BLAZE_CONSTRAINT_MUST_BE_BLAS_COMPATIBLE_TYPE( ElementType_<MT3> );
  
     using ET1 = ElementType_<MT1>;
  
     const size_t m( (~A).rows() );
     const size_t n( (~A).columns() );
     const size_t mindim( min( m, n ) );
  
     if( ( !IsResizable<MT2>::value && ( (~L).rows() != m || (~L).columns() != mindim ) ) ||
         ( !IsResizable<MT3>::value && ( (~Q).rows() != mindim || (~Q).columns() != n ) ) ) {
        BLAZE_THROW_INVALID_ARGUMENT( "Dimensions of fixed size matrix do not match" );
     }
  
     if( IsSquare<MT2>::value && m != mindim ) {
        BLAZE_THROW_INVALID_ARGUMENT( "Square matrix cannot be resized to m-by-min(m,n)" );
     }
  
     const std::unique_ptr<ET1[]> tau( new ET1[mindim] );
     DerestrictTrait_<MT3> l( derestrict( ~L ) );
  
     if( m < n )
     {
        (~Q) = A;
        gelqf( ~Q, tau.get() );
  
        resize( ~L, m, m, false );
        reset( l );
  
        for( size_t i=0UL; i<m; ++i ) {
           for( size_t j=0UL; j<min(i+1UL,n); ++j ) {
              l(i,j) = (~Q)(i,j);
           }
        }
  
        lq_backend( ~Q, tau.get() );
     }
     else
     {
        l = A;
        gelqf( l, tau.get() );
        (~Q) = submatrix( l, 0UL, 0UL, n, n );
        lq_backend( ~Q, tau.get() );
  
        for( size_t i=0UL; i<m; ++i ) {
           for( size_t j=i+1UL; j<n; ++j ) {
              reset( l(i,j) );
           }
        }
     }
  }
  //*************************************************************************************************
  
  } // namespace blaze
  
  #endif
  